A closed-topped cylindrical tin can is to be made with a volume of
25 cm ^3.
a) Determine a function to model the height h, of a can in terms of the radius, r.
b) Determine a function to model the surface area A, of the can in terms of the radius, r.
c) What is an appropriate domain for the surface area function, A(r).
pi r^2 h = 25
so
h = 25/(pi r^2)
A = 2 pi r^2 + 2 pi r h
but we know h(r)
A = 2 pi r^2 + 2 pi r * 25/(pi r^2)
A = 2 pi r^2 + 50/r
r better be >0
a) To determine a function to model the height h of a cylindrical tin can in terms of the radius r, we can use the formula for the volume of a cylinder. The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height.
In this case, the volume is given as 25 cm^3. Thus, the equation becomes:
25 = πr^2h
To find the function to model the height, you can isolate 'h' by dividing both sides of the equation by πr^2:
h = 25 / (πr^2)
Therefore, the function to model the height h in terms of the radius r is h = 25 / (πr^2).
b) To find a function to model the surface area A of the can in terms of the radius r, we need to find the formula for the surface area of the cylinder. The formula for the surface area of a cylinder is A = 2πrh + 2πr^2, where A is the surface area, r is the radius, and h is the height.
Replacing the function for the height h from part a, the equation becomes:
A = 2πr(25 / (πr^2)) + 2πr^2
Simplifying further:
A = 50 / r + 2πr^2
Therefore, the function to model the surface area A in terms of the radius r is A = 50 / r + 2πr^2.
c) An appropriate domain for the surface area function, A(r), would be the set of positive real numbers, excluding zero. Since radius cannot be negative or zero in this context, the domain for A(r) would be all positive real numbers (r > 0).